Meanwhile, the contribution \(\Delta \mathcal {L}\)  in \(\mathcal {L}_{\text{5D}}\)  is a total derivative we add for convenience which generalizes a total derivative from our unstabilized analysis [1]}, [2]}. It cancels terms in the Lagrangian at linear order, eliminates mixing between tensor and scalar 5D fields at quadratic order, and simplifies the vertices relevant to this paper.If such a term were not introduced here, we would recover its effects (at least to quadratic order) as additional total derivative terms needed to make the Lagrangian explicitly canonical. This \(\Delta \mathcal {L}\)  naively generalized the \(\Delta \mathcal {L}\)  introduced in refs. [1]}, [2]}, and (unlike the latter) does not eliminate Dirac deltas or twice-differentiated quantities to all orders. Explicitly, we define it as
\(\Delta \mathcal {L} \equiv \dfrac{2}{\kappa ^2} \partial _y\left[\dfrac{w^2}{v} \sqrt{-\text{det}g}\left( -3\frac{\partial _y(w)}{w}\right)\right] ~. \) 
